Peel the mangos and cut each into two halves (total of 12 pieces). In a heavy 14-inch sauté pan, melt the butter, sugar, and rum together over high heat, stirring occasionally with a wooden spoon. Place mango halves round side down into the sugar mixture. Continue to cook over high heat for approximately 3 minutes, or until the syrup is a light straw color.

Arrange the mango halves, round side down, on a non-stick, half-sheet baking pan. Spoon the syrup over the mangos. Place one piece of pastry on top of each mango. To allow the pastry to bake evenly, bake for 15 minutes, then rotate the baking sheet and bake for another 15 minutes. With a spatula, carefully lift the tarts, one at a time, and flip onto a serving platter or onto individual dessert plates, mango side will be facing up. Drizzle remaining syrup from the baking sheet over the tarts.

Crème Anglaise: In a 2-quart pot off the stove, whisk together egg yolk, sugar, and salt.

Heat the coconut milk in a 1-quart pot until it reaches a simmer. Whisk coconut milk into the egg yolk mixture and place it on the stove over medium heat. Stir constantly with the rubber spatula while scraping the bottom and sides of the pot to prevent any of the egg from overheating. Cook for approximately 4 minutes or just until the mixture has thickened enough to coat a wooden spoon evenly.

Strain over the bowl and add in the vanilla and rum. Spoon generously over the tarts or just around the base of the tarts.